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Theorem rspcdvinvd 40402
Description: If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
rspcdvinvd.1 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
rspcdvinvd.2 (𝜑𝐴𝐵)
rspcdvinvd.3 (𝜑 → ∀𝑥𝐵 𝜓)
Assertion
Ref Expression
rspcdvinvd (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdvinvd
StepHypRef Expression
1 rspcdvinvd.3 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
2 rspcdvinvd.2 . . 3 (𝜑𝐴𝐵)
3 rspcdvinvd.1 . . 3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
42, 3rspcdv 3612 . 2 (𝜑 → (∀𝑥𝐵 𝜓𝜒))
51, 4mpd 15 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890  df-ral 3140
This theorem is referenced by:  imo72b2  40403
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